Geometrically, this is the set of points z such that the distance of z from z = 3 plus the distance of z from z = -3 is equal to 10. Geometrically this defines an ellipse. The points z = 3 and z = -3 are the focal points. The length of the major axis is 10. By symmetry the centre is at (0, 0).
Therefore, if z = x + iy, then the Cartesian equation has the form .
To get b, you can apply Pythagoras Theorem:
Since the focii are at (3, 0) and (-3, 0), .
Therefore the Cartesian equation is .
Of course, you can do it the hard way by substituting z = x + iy into and going through the algebra. If I have time later I'll post the highlights.